Application of Simple Adaptive Control to a DC/DC Boost Converter with Load Variation

Transcription

1 Application of Simple Adaptive Control to a DC/DC Boost Converter with oad Variation Goo-Jong Jeong 1, In-Hyuk Kim 1 and Young-Ik Son 1 1 NPTC, Department of Electrical Engineering, Myongji University, Yongin, Korea (Tel : ; (gold85, hyucin, sonyi)@mju.ac.kr) Abstract: Output voltage of a DC/DC power converter is likely to be distorted if variable loads exist in the output terminal. Unlike the buck type converter case, the regulation problem of the boost (step-up) converter is particularly difficult because the system is non-minimum phase with respect to the output voltage to be regulated. This paper presents a simple adaptive control (SAC) approach to maintain a robust performance against load variation of a boost DC/DC power converter system. In order to apply the SAC algorithm the transfer function of the controlled plant should be almost strictly positive real (ASPR). Since the converter system does not possess ASPR, a parallel feedforward compensator (PFC) is needed to implement the SAC algorithm. We first design a PI controller and make a series connection with the linearized model of the power converter at the operating point. Then a stabilizing PD controller is constructed for the connected system. Inverse of the obtained PD controller is used for the PFC that renders the parallel-connected system to be ASPR. Computer simulations show the effectiveness of the proposed control algorithm. Keywords: DC/DC Boost Converter, Simple Adaptive Control (SAC), Non-minimum Phase System. 1. INTRODUCTION As the amount of fossil fuels decreases, it is necessary to explore other solutions for electric power energy generation. Developed and developing countries around the world are looking to alternative energy as part of the efforts to reduce their dependence on crude oil and to enhance energy security. Renewable energy sources such as fuel cell and photovoltaic have attracted many researchers [1, 2]. Since the DC voltage generated by fuel cell or photovoltaic varies widely in magnitude, DC/DC boost (stepup) conversion stage is essential to provide highly regulated DC voltage. In the process of the voltage step-up, the unexpected transient frequently happens owing to uncertain load variations. This demands extensive research on the robust control for the boost converters along with the study on renewable energy [3 5]. Because the boost converter exhibits highly nonlinear and non-minimum phase properties with uncertain loads, it is imperative to design a robust controller for compensating the load perturbation. There have been continuous efforts to design control strategies to improve the performance of the power converter. (See the references [6 16] and therein.) SAC (simple adaptive control) is a simple and robust algorithm for unmodeled dynamics since the order of reference model can be chosen almost freely regardless of that of controlled system. Because of these features the SAC has been implemented successfully in diverse applications such as flexible structures, power system, robotics, motor control and other [17, 18, 21]. However, in order to apply SAC algorithm, the transfer function of the controlled plant should be almost strictly positive real (ASPR). It has been shown that plants with minimumphase transfer functions of relative degree one are ASPR. This paper concentrates on the control of the DC/DC boost converter with load variation. Among the various robust control techniques the SAC is considered. Owing to the non-minimum phase zero the boost converter is not ASPR. Hence the SAC cannot be directly applied to design of a robust control for the system. This paper presents a way of constructing a parallel feedforward compensator (PFC) with which the augmented system can satisfy the ASPR condition [18 21]. By doing this we can make use of the SAC algorithm to design a robust control for the converter. This is the first result on the application of the SAC to DC/DC boost converters. This paper is organized as follows. Section 2 introduces a mathematical model of a DC/DC boost converter and the SAC. In Section 3 computer simulations confirm the performance of the SAC-based control for the DC/DC boost converter against load perturbation. We have compared the closed-loop performance with three different controllers, designed using 1 a PI controller combined with a PI observer-based disturbance observer [22, 23]; 2 the integral state feedback control; 3 the proposed control. Finally, conclusions are drawn in Section SAC FOR DC/DC BOOST CONVERTER 2.1 Boost converter model DC/DC power converters are the devices that change a certain electrical voltage to another level of voltage by switching action. This paper considers the DC/DC boost converter as shown in Fig. 1. The switched DC/DC converter is composed of two functional blocks which are the power stage and the switch controller stage. The switch is controlled by the pulse width modulator (PWM). When the switch is turned on, Fig. 1 is divided into two independent circuits. The current through the inductor increases and the energy is stored in it. If the switch is turned off, the stored energy in the inductor decreases and the polarity of the voltage changes so

2 modeling method is used to obtain the following model. [ R ẋ = 1 D ] [ Vc ] 1 D C 1 x + R C I u C y = [ 1 ] (3) x Fig. 1 Diagram of the Boost Converter Control System that the inductor voltage is added to the input voltage. Hence, both the voltage across the inductor and the input voltage V i charge the output capacitor. By the repetitive operation explained above the circuit of Fig. 1 makes the output voltage higher than the voltage source level [24]. By using an average switching method the mathematical model of Fig. 1 is described by di dt = R i 1 (1 d)v c + 1 V i dv c dt = 1 C (1 d)i 1 R C v c (1) where i is the inductor current; v c is the output voltage; V i is the input voltage source; control input d is the duty ratio ( d < 1); R, and C are the circuit parameters; R is the load resistance. When the inductor resistance is disregarded (i.e. R = ) from model (1), the steadystate relation between the voltage source and the output voltage is given by V c = 1 1 D V i (2) where V c is the output voltage and D is the duty ratio at the equilibrium point. This equation shows that the output voltage is always higher than the input voltage if the system is stable at the range of duty ratio [24]. As we can see in (1) the boost converter is a bilinear system including the multiplication of input and system states [6]. And it is a non-minimum phase system because the zero dynamics of the system is unstable as to the output voltage [14]. This property makes the control problem of the converter very complicated and many works have been made by the researchers in the field of control theory as well as power electronics for many years e.g. [6 16]. Next section presents a new controller to enhance the robustness against the perturbation of output voltage caused by uncertain load resistance R. The proposed controller is designed via SAC (Simple Adaptive Control) approach [17]. In order to apply the SAC approach we consider the linearized model of the power converter system at the desired equilibrium point. The small signal where x is the system states [ i I v c V c ] T and the input u is d D. The transfer function of (3) shows that its zero is located at z 1 = V c(1 D)/I R. Since the inductor resistance R is very small, the zero is at the right half plane. This implies the linearized system (3) is also non-minimum phase. 2.2 Simple Adaptive Control (SAC) Algorithm Since it is a simple and robust algorithm for unmodeled dynamics, the SAC has been implemented successfully in diverse applications (see e.g. [18]). In the SAC algorithm the plant is described as ẋ p = A p x p + B p u p y p = C p x p. (4) The plant output y p is required to asymptotically track the output of the following model y m. ẋ m = A m x m + B m u m y m = C m x m (5) Therefore the objective is to find the control input u p such that the plant output y p approximates the output y m. If we let e y = y m y p, the output of the SAC is defined as u p = K(t)r(t) (6) where r(t) = [e y x m u m ] T ; K(t) = K I (t) + K P (t); K I (t) = [ K Ie (t) K Ix (t) K Iu (t) ]; (7) K I (t) = e y r T Γ I [ σk Ie ]; (8) K P (t) = e y r T Γ P ; (9) and σ is a small positive constant; Γ I and Γ P are constant matrices [17, 21]. In order to prove the stability of the closed-loop system the controlled plant is required to be almost strictly passive (ASP) and its transfer function almost strictly positive real (ASPR). If the given plant is not ASPR, a parallel feedforward compensator (PFC) can be used to satisfy the ASPR condition [18 21]. This paper proposes an SACbased adaptive output feedback control algorithm based on Fig. 2 [21]. Since the boost converter is a non-minimum phase system, the SAC cannot be directly applied to the system. A PFC should be developed to make the augmented system minimum phase and have relative degree one. In this case the augmented output y a in Fig. 2 is forced to asymptotically track the model output y m. Hence it is important to guarantee that the PFC output y s is sufficiently small

3 so that the actual plant output y p is almost same as the augmented output y a. The design of a PFC for the converter system consists of two steps. We first design the TI Controller in Fig. 2 that stabilizes the linearized model of the power converter (3) at the operating point. (In the next section a PI controller is designed for this system.) Then a stabilizing PD controller is constructed for the series connection of the T I Controller and (3). Inverse transfer function of the obtained PD controller is used for the PFC in Fig. 2. The PFC renders the augmented system to be almost strictly positive real (ASPR). 3. DESIGN EXAMPE The performance of the proposed controller is tested by computer simulations with SimPower system of Matlab/Simulink. Comparisons have been made with another controllers: 1 the conventional PI controller with a PI observer-based disturbance observer (PI DOB) and 2 an integral state feedback controller with a PI DOB [22, 23]. As a robust control algorithm the PI DOB is employed to reduce the effect of disturbances (such as load changes in this paper) [22]. The system parameters of the boost converter are determined as in Table 1. Table 1 Simulation parameters Circuit Parameters Inductor = 1mH Inductor Resistance R =.6Ω Capacitor C = 1mF Voltage Source V i = 12V Desired Output Voltage V o = 24V oad Resistance R = 5Ω (Perturbance) 14.28Ω (5msec) 5Ω (.1sec) Nominal Equilibrium Point Output Voltage V c = 24V Inductor Current I = 1.111A Duty Ratio D =.5253 In order to design a robust controller via SAC approach, the linearized model (3) is considered. The transfer function of the linearized model (3) is given by G(s) = 111s + ( ) s s + ( ). (1) Notice that the transfer function is non-minimum phase. As mentioned before, the SAC cannot be directly applied to the boost converter because it is non-aspr. We consider the control structure in Fig. 2. The first step to design the PFC is to find the TI controller for (1). A PI controller is designed as follows: C 1 (s) =.1s +.3 s (11) Next we design the PFC that makes the augmented plant minimum phase and relative degree one. For the purpose the series connection C 1 (s)g(s) is considered. A PD controller C 2 (s) is constructed such that C 2 (s) stabilizes the series connection C 1 (s)g(s) 1. Since the zeros of 1 + C 1 GC 2 (s) are equivalent to those of (C 1 G + )(s), the PFC is designed with the inverse of the C 2 (s) i.e. PFC = C2 1 (s). Finally the inverse of a stabilizing PD controller (C2 1 (s)) is chosen as C 1 2 PFC = C2 1.1 (s) =.1s + 1. (12) The transfer function of C 1 G(s) + PFC is given by C 1 G(s) + PFC =.8989s s 2 +( )s+( ) s s 3 +( )s 2 +( )s. (13) Figure 3 shows the root-locus of (13). The root locus verifies that the PFC makes the augmented system minimum phase and relative degree one. Since the (C 1 G + PFC) is ASRP, the SAC approach can be tried to achieve the robust performance. Imaginary Axis Root ocus Real Axis Fig. 3 Root-locus for C 1 G(s) + PFC Figure 4 compare the closed-loop performance with three different controllers: 1 a PI controller combined with a PI observer-based disturbance observer [22, 23]; 2 the integral state feedback control; 3 the proposed control. In the simulation the model equation for the SAC is ẋ m = 3x m + 3u m y m = x m. (14) The constant σ =.1; Γ I = diag([1, 1, 1]) and Γ P = diag([,, 1]) were used. The load resistance R has been changed from 5Ω to 14.28Ω at 5ms. The value changes back to 5Ω at.1sec. Figure 4 shows that the proposed SAC-based control has an advantage over the other control laws. Unlike the integral state feedback control the proposed control does require the current measurement. 1 This means that the closed-loop system C 1GC 2 (s) is stable. 1+C 1 GC 2 (s)

4 Fig. 2 Diagram of SAC Using PFC and TI Controller [21] 4. CONCUSION Output Voltage This paper has presented an SAC (simple adaptive control)-based output feedback controller to improve the dynamic response of the DC/DC boost converter against the load variation. Since the boost converter is non- ASPR, we have designed an TI controller and a PFC for the linearized model of the converter system. The simulation result shows that the proposed control ensures the regulation of output voltage effectively under the load uncertainty. 1 5 PI Controller with PI DOB Integral State Feedback with PI DOB Proposed Controller Using SAC (a) Output Voltage v c Inductor Current PI Controller with PI DOB Integral State Feedback with PI DOB Proposed Controller Using SAC (b) Inductor Current i Fig. 4 Simulation Results for oad Variation ACKNOWEDGEMENT This work was supported by the 2nd Brain Korea 21 Project. This work was supported by the ERC program of MOST/KOSEF (Next-generation Power Technology Center). REFERENCES [1] J. Wang, F.Z. Peng, J. Anderson, A. Joseph, R. Buffenbarger, ow Cost Fuel Cell Converter System for Residential Power Generation, IEEE Trans. on Power Electronics, Vol. 19, No. 5, pp , Sep. 24. [2] J.H. ee, H.S. Bae, B.H. Cho, Resistive Control for a Photovoltaic Battery Charging System Using a Microcontroller, IEEE Trans. on Industrial Electronics, Vol. 55, No. 7, pp , Jul. 28. [3] R. Wai, W. Wang, C. in, High-Performance Stand-Alone Photovoltaic Generation System, IEEE Trans. on Industrial Electronics, Vol. 55, No. 1, pp , Jan. 28. [4] M.H. Todorovic,. Palma, N. Enjeti, Design of a Wide Input Range DC-DC Converter with a Robust Power Control Scheme Suitable for Fuel Cell Power Conversion, IEEE Trans. on Industrial Electronics, Vol. 55, No. 3, pp , Mar. 28. [5] Y. Chen, K. Smedley, Three-Phase Boost-Type Grid-Connected Inverter, IEEE Trans. on Power Electronics, Vol. 23, No. 5, pp , Sep. 28.

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